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Topological statistics, in the form of persistence diagrams, are a class of shape descriptors that capture global structural information in data. The mapping from data structures to persistence diagrams is almost everywhere differentiable, allowing for topological gradients to be backpropagated to ordinary gradients. However, as a method for optimizing a topological functional, this backpropagation method is expensive, unstable, and produces very fragile optima. Our contribution is to introduce a novel backpropagation scheme that is significantly faster, more stable, and produces more robust optima. Moreover, this scheme can also be used to produce a stable visualization of dots in a persistence diagram as a distribution over critical, and near-critical, simplices in the data structure.

The availability of e-Inc reading devices, such as Amazon's Kindle, and tablets like the Apple's iPad dramatically pushed the market for electronic books and magazines. The well known advantages of electronic content compared to printed media are instant delivery, multimedia capability, less weight and additional media related services, such as dictionaries and language translation. Beside these known advantages, mobile electronic devices further offer a broad spectrum of sensors and input/output capabilities to further enrich the presentation of modern electronic media. Within this work we propose a system architecture and a mobile reader software for further enhance the capabilities of ebooks and magazines. The proposed enhancement enables ebooks to react context- and situation-aware onto the readers demands in various application domains. By adding active elements, such as rules and conditional events to electronic media, we enable smart behaviour of electronic content. Furthermore it enables the reader complex interactions with an ebook, which goes further than just page flipping and chapter selection. A case study presents a personalized travel guide which reacts context-sensitive according to the user's actual situation.